Constraint efficient firm entry and the bargaining power

W(¯x^S)−W(¯x) W(¯x) (%)

∆(1) ∆(2) ∆(3) ∆(4) ∆(5) ∆(6) ∆(7) ∆(8)

≈ −0.56 ≈ −12.597 ≈0.135 -0.08 8.4 -2.63 -6.25 17.4

∆(9) ∆(10) ∆(11) ∆(12) ∆(13) ∆(14) ∆(15) Total

11.7 - - - - -2.87 -2.73 ≈9.9

Table 3.6: Numerical decomposition of the welfare gain with free-entry

displays the equilibrium promotion cutoff as a function ofK(left panel) and the corresponding equilibrium stock of firms (right panel). As expected, the number of active firms declines as the entry cost increases. Considering the effect of higherKon the optimal promotion timing, then there are several effects. Firstly, higher entry cost means that the present value of an entering firm must also increase. In order to achieve that firms must earn higher profits.

Assuming market conditions remain constant otherwise and under fixed match output sharing rule, this is possible only if the average output per match is increased. Hence, firms have to let their junior workers accumulate more experience and delay internal promotions.

Secondly, there is a labour demand effect coming from decreasing firm competition as the equilibrium number of firms declines. Since there are less competing vacancies in both sub-markets, it is easier to fill an open vacancy and bothq₁ and q₂ go up (see figure 3.19 in Appendix B). The effect of a simultaneous increase of both of those variables on ¯xis, however, ambiguous since they have an opposite effect on the optimal promotion timing. Higher junior vacancy-filling rate is associated with earlier promotions while higher senior vacancy-filling rate leads to later internal promotions⁴. Overall, the effect of lower competition in the senior market dominates in this setting and optimal promotion timing rises in response to higher entry cost.

Similarly to the decentralized equilibrium, the socially optimal ¯x^S increases in K. As it will be discussed below, if the condition are relatively favourable for firms, implying that many firms can stay active on the market, the social planner can maximize welfare by choosing immediate or very fast promotion and employing many workers at the high productivity senior jobs. If, however, there are few active firms because of unfavourable market conditions, such as in this case, a high entry cost, welfare is maximized by delaying promotions and increasing average match output. In the next section we explore the relationship between firm entry and the socially optimal promotion timing in more detail.

Figure 3.12: Left panel: Optimal ¯xand ¯x^S as a function of entry cost. Right panel: Number of firms under decentralized and socially optimal equilibrium with free-entry.

Increasingβhas several effects on firms’ promotion decisions. On the one hand, if a higher share of the output goes to workers, firms could compensate by delaying promotions and thus increasing output per worker. Also, as β increases, firms profits decline and fewer firms are able to stay active. Hence, many workers compete for few vacancies and the vacancy-filling rates increase. As discussed above, a simultaneous increase in both q₁ and q₂ has an ambiguous effect on ¯x. We see that for low to middling values of β the effect of decreasing firm competition on the senior market dominates, so firms delay promotions. For higher values of β, the equilibrium number of firms declines so much that the effect on promotions is reversed, i.e. increasingβ is associated with a decrease in promotion timing. The stock of firms at β = 0.75 is nF = 0.261 which is more than twice less than in the case β = 0.5 and for β > 0.75, it approaches 0, so ¯x cannot be computed. For such high values of workers’

bargaining power, there are even fewer firms and potential vacancies and many competing searching workers in the market. Consequently, the job-filling rates increase steeply in β (see figure 3.20 in Appendix B). Also, we observe that q₁ increases faster compared to q₂. This combined with the decreasing senior job-finding rate λ2 becomes the dominant effect and so for high values of workers’ bargaining power, promotion timing in the decentralized equilibrium decreases.

Next, note that the socially efficient promotion timing also depends on β. The red curve in figure 3.13 plots ¯x^S for different values ofβ. If β = 0, then the social planner maximizes the stock ofe₀ workers given the matching frictions and the free-entry condition. This is the extreme case in which firms retain all of the output from the match and is not of interest for the analysis. For β ∈ [0.3,0.4] immediate promotion is optimal. Since firms retain a larger share of the total output, many firms enter the market, there is a high firm competition in both submarkets and the vacancy-filling rates are low. By choosing immediate promotions the social planner is thus able to employ many workers at the high productivity senior positions.

Even though increasing ¯x also translates into higher output once the workers are at the e2 level, the firm competition effect dominates here. Increasing ¯x in this case reduces q2

Figure 3.13: Optimal promotion and the bargaining power.

even further which suppresses the profits of firms and drives some firms out of the market.

Hence, setting higher promotion requirement acts as a barrier to entry for new firms and the equilibrium stock of firms monotonically decreases. To illustrate this figure 3.21 in Appendix B displays the social planner’s objective function, the firm stock and the job-filling rates for varying ¯x and β = 0.35.

At β = 0.5, we have ¯x^S = 7.4 which is the case depicted in figure 3.10. As β increases further, so does the socially optimal promotion timing. Figure 3.22 in Appendix B display the caseβ = 0.75. Comparing it to the case β = 0.35, here the stock of firms is much lower for all considered promotion cutoffs. This implies that firm competition is lower and the the vacancy-filling rates are higher (right panel of figure 3.22). Setting a higher promotion requirement then increases the average output per match which leads to higher firm profits and consequently higher entry up to a certain value of ¯x. Hence, here the productivity effect dominates and welfare rises as promotions increase up from ¯x= 0. Choosing too high promotion requirement, however, suppresses q₂ drastically so it becomes difficult for firms to fill their high productivity senior jobs and again welfare starts to decline.

These considerations imply that the way firms and workers split the match output is crucial in determining the social efficiency of firm’s promotion timing and entry. To explore this relationship deeper, we next assume that β1 is the share of output that goes to junior workers andβ₂ is the share of output that accrues to senior workers. In what follows, the aim is to answer the question whether there is a pair{β1, β2}under which the socially optimal and the decentralized equilibrium coincide. Firstly, assumingβ₂= 0.5 is constant, then the firm’s promotion choice is decreasing in β₁: ∂x/∂β¯ ₁ <0 (see right panel of figure 3.13, blue curve) which is similar to the fixed entry case. Further, for fixedβ1 = 0.5,∂¯x/∂β2 is non-monotone mirroring the overall relationship betweenβ and ¯x (see right panel of figure 3.13, red curve).

In the following step, we plot the decentralized and socially optimal values for the promotion timing against the bargaining power of the junior worker for fixed bargaining power of senior workers. The result is shown in figure 3.14 which plots the casesβ₂ = 0.35,β₂= 0.38β₂ = 0.4 (top row, left to right) and β2 = 0.5, β2 = 0.7 β2 = 0.8 (bottom row, left to right) . The

Figure 3.14: Comparative statics of decentralized and socially optimal promotion cutoffs ¯x,

¯

x^S with respect to β1 for fixed β2. Top row: left panel: β2 = 0.35, middle panel: β2 = 0.38, right panel: β2 = 0.4.Bottom row: left panel: β2 = 0.5, middle panel: β2 = 0.7, right panel:

β₂ = 0.8

benchmark case: β2 = 0.5 is depicted in the left panel, bottom row of the figure. We see that there is no value of β1 for which the social planner’s promotion timing coincides with the decentralized equilibrium if β₂ = 0.5. Also, ¯x is decreasing in the bargaining power of junior workersβ1, while ¯x^S is increasing in it for low to middling values ofβ2 (top row of the figure). Since, the social planner maximizes the total wage bill together with the output of workers in simple jobs, it follows that if the share of output that workers retain at a certain hierarchical level increases, welfare can be improved if more workers are employed in those jobs. So here, sinceβ₂ is fixed, increasingβ₁ leads to later promotions. Firms have, however, the opposite response, such that ¯x decreases in β₁. This is the case because for higher β₁ it becomes less profitable to retain a worker at the junior level so firms choose earlier promotion timing.

Furthermore, increasing β₂ we see that the ¯x curve shifts outwards. If senior workers receive a larger share of the match output, firms optimally slow down promotions for all plausible values of β₁. Qualitatively, the socially optimal promotion timing still increases inβ₁ for lower values of β₂ (see top row of figure 3.14). The intercept and the slope of ¯x^S, however, vary greatly. Forβ2 = 0.38, for instance, immediate promotions are socially optimal for low values ofβ₁ (middle panel, top row). The intuition behind is similar to above: since bothβ₁ andβ₂ are relatively low, welfare is maximized by setting immediate promotions and employing more workers in senior jobs. As β1 increases, however, the ¯x^S curve begins to increase steeply. The numerical simulations show that there exists a pair (β₁, β₂) such that β1 > α1andβ2< α2for which the decentralized and the socially optimal equilibrium coincide.

Moreover, the pair is not unique. For instance, at (β₁, β₂) = (≈0.82,0.35) and (β₁, β₂) = (≈

0.92,0.38), depicted in the left and middle top row panels of figure 3.14, the socially optimal and decentralized equilibrium coincide. In the first case: (β1, β2) = (≈0.82,0.35), we have

¯

x= ¯x^S ≈26.5, while for (β₁, β₂) = (≈0.92,0.38) it follows: ¯x= ¯x^S ≈28. Out of those two, however, only the equilibrium corresponding to the pair (β1, β2) = (≈ 0.82,0.35) is stable.

The optimal response functions for the two cases are displayed in figure 3.24 in Appendix B.

For higher values ofβ₂(middle and right panel of bottom row of figure 3.14) the promotion timing that firms choose is still decreasing inβ1. The socially efficient promotion cutoff, on the other hand, exhibits a non-monotone relationship with the bargaining power of junior workers, such that it is increasing at first and starts to decline for higher values of β₁. Furthermore, the intercept of the curve increases with higher β2 such that immediate promotion is not efficient for any value ofβ1. The reason is similar to the one discussed above for the case that workers on both hierarchical levels have the same bargaining power. Further, when both β₁ andβ2 are high, ¯x^S declines inβ1 since the number of active firms approaches 0. Figure 3.23 in Appendix 3.7 plots ¯x and ¯x^S forβ2 = 0.9. We see that in this case the slope of∂x¯^S/∂β1 is negative over the whole range where the socially optimal and decentralized equilibrium could be computed. Very few firms are active in the market if β2 = 0.9, so increasing β1 leads to decreasing promotion timing as the social planner maximizes the wage bill given the very few employment opportunities. However, this indicates that if β₂ ≥α₂, the socially optimal and decentralized equilibrium never coincide.

The above discussion highlights that the share of output accruing to senior workers is above the value needed so that the socially optimal and decentralized equilibrium coincide, while the opposite is true for the share of output earned by junior workers. This contributes to firms’ incentives to delay promotions inefficiently long which leads to under-entry of firms and a stock of e₀ workers which is above the socially efficient level. This reveals that firms are not adequately compensated for creating the high productivitye2 jobs and therefore firm entry is biased downwards. High wages in the senior market suppress firm creation which implies that the optimal bargaining power in that market has to be below the traditional Hosios value. Moreover, the adverse effect of strategic complementary can be neutralized if the bargaining power of workers in the junior market is set above the Hosios value as to deter firms from delaying promotion inefficiently long. Under those two conditions we can find multiple equilibria for which the decentralized equilibrium is also constrained efficient.

The effects on workers in different hierarchical levels, however, are diverse. If we consider the case (β₁, β₂) = (≈ 0.82,0.35), then the resulting vector of transition rates is: {λ₁ = 0.17, λ2 = 0.16, q1 = 0.03, q2 = 0.03}. Comparing these values to the ones in table 3.5 implies that workers in simple jobs gain from having a higher job-finding rate compared to the decentralized equilibrium in the case (β₁, β₂) = (0.5,0.5). Workers who are searching for a senior job are in a less favourable position since their job-finding rate has decreased slightly.

Finally, the wage gain associated with being promoted, either internally or via changing firms is smaller due to the fact that β1 has increased andβ2 is lower.

Figure 3.15: Left panel: optimal sharing rule under fixed ¯x= 40 andy0=d1+c1. Maximum is achieved at φ = 0.678 Right panel: optimal sharing rule under fixed ¯x = 40 and y0 = 0.

Maximum is achieved atφ= 0.619